Before we really start

Why using `R` for genetics/genomics?

you will need to learn R anyhow to analyse non genetic data
you may not want to learn how to use 50 different pieces of software with various syntax
you may want to integrate your analysis into a larger workflow (e.g. statistical analysis, plots)

Outcome

gain familiarity with R
understand how microsatellite data can be handled in R
perform basic genetic analyses using the packages adegenet, poppr and pegas
use R Markdown to produce a report of the analyses, incorporating the workflow and results

Homework

analyse and interpret a Eurasian lynx dataset & produce a report (PDF) as part of course evaluation

About this document

This document is an R Markdown document. An R Markdown document is the combination of a Markdown document with chunks of R code in the middle that are evaluated using an R package called knitr.

R Markdown document can be used interactively or they can be turned into beautiful HTML document or other (e.g. PDF, Word, …) if you click on the button “Knit” on top.

The benefits of using an R Markdown document is that you have your text, scripts, results, plots and tables in a single place.

You can edit the current R Markdown file to add your own notes to it (but save it under a different name, so that you can go back to the original version if you need to) and to try out or even modify the R code.

If you prefer to see pure R code only, just do:

#file.remove("markdown_R4G3_course.R")
knitr::purl("markdown_R4G3_course.Rmd")

Note: it will create a file called markdown_R4G3_course.R but it won’t overwrite it if it already exists. So, if you want some changes to be taken into account, make sure you delete the R file beforehand (e.g., by running the first line in the chunk above).

Basic Markdown syntax

Just write plain text in the script. You can use different number of underscores to indicate:

italic (1 underscore on each side)
bold (2 underscores on each side)
both (3 underscores on each side)

For example _word_ is rendered as word.

Basic `knitr` syntax

Use ```{r} to start an R chunck and ``` to close it.

For example if you type 1 + 1 within a chunk, it will be displayed as:

1 + 1

## [1] 2

Then, to evaluate the chunck, just press CTRL-R after having put your mouse cursor inside the chunck (anywhere).

You can learn more about R Markdown on this online book or if you already know a little, the R Studio cheatsheet will help you to keep it all in mind.

Getting all the `R` packages ready

We will use the R packages adegenet, poppr and pegas for the analyses and ggplot2, lattice and viridisLite for the plots. If you have already installed before, you can simply load the packages as follow:

library(adegenet)
library(poppr)
library(pegas)
library(ggplot2)
library(lattice)
library(viridisLite)

Note 1: If one of the package is missing, then you must install it BEFORE loading them. So you may need to use, for ex:

install.packages("adegenet")

Note 2: To be able to “knit” the R Markdown, you will also need other R packages but try kniting and R Studio should do the rest for you.

We also add a few extra functions we coded for you:

source("scripts/tools.R")

Importing the data into `R`

The `genepop` format

We will be using a very common type of input file for microsatellite data, which was originally developed for a stand-alone program called Genepop. You will learn how to create a genepop file with Jörns tomorrow. Today we will use one that we have made for you.

Here is the basic structure:

comment line
          locus-1, locus-2, locus-3, ... 
pop
Ind-1  ,   100102   135135   204208  ...
Ind-2  ,   100100   131139   200208  ...
Ind-3  ,   102102   131139   200204  ...
Ind-4  ,   000000   135139   208208  ...
...

the first line is a “comment line” - you can keep notes on your project here. E.g. “24 animals, 8 loci”
the second line contains the names of the loci (locus-1, etc).
the third line has a “pop flag” that indicates the next samples belong to the same population.
below this (until the next pop flag), every line is a multi-locus genotype per individual belonging to the population
- the first column with sample ID (followed by a comma)
- the following columns with the genotype at a given locus (the two alleles, without a separator)
- missing data at a locus is denoted by “000000”

Everyone comfortable with the terms locus, allele and genotype?

Reading your data into `R`

In the folder data in this R Studio project you can find the genepop file called “nancycats.gen”.

We will create an R object that contains the data in nancycats.gen using the read.genepop() function of adegenet. Here we need to provide the path of the file as "/data/nancycats.gen" using file =, and some information about how microsatellite data is encoded in the file using ncode =. (And here we also set the argument quiet = but you don’t have to, this is just to shorten this document by not displaying some messages.)

myData <- read.genepop(file = "data/nancycats.gen", ncode = 3)

## 
##  Converting data from a Genepop .gen file to a genind object... 
## 
## 
## File description:  nancycats dataset from adegenet (incl. mods)  
## 
## ...done.

Why did we use ncode = 3 ?

Now the data is read into R.

If there had been a problem with the data file (e.g. incorrectly formatted), then we would have gotten an error message. Let’s try:

badFormat <- read.genepop(file = "data/badFormatting.gen", ncode = 3, quiet = TRUE)

## Error in dimnames(x) <- dn: length of 'dimnames' [1] not equal to array extent

We “broke” this input file by including fewer locus names than data. The error message is telling us that the length of dimnames [1] (= number of locus names) does not correspond to the number of columns containing the data.

As you can see, we are lucky that R provides us a clue about where the problem is.

Another way we can generate an error is by providing wrong arguments to the function, despite having a properly formatted file:

badInput <- read.genepop(file = "data/nancycats.gen", ncode = 2, quiet = TRUE)

## Error in read.genepop(file = "data/nancycats.gen", ncode = 2, quiet = TRUE): some alleles are not encoded with 2 characters
## Check 'ncode' argument

Unsurprisingly, there will be an error if you type the path to the file incorrectly.

One way to limit such issues is to list all genepop files in the folder containing the data (to make sure that the file is where you are trying to read it):

dir(path = "data/", pattern = "*.gen")

## [1] "badFormatting.gen" "lynx.gen"          "nancycats.gen"

`R` representations of your data

The `genind` format

We haven’t looked at our in data yet!

myData

## /// GENIND OBJECT /////////
## 
##  // 133 individuals; 9 loci; 104 alleles; size: 85.2 Kb
## 
##  // Basic content
##    @tab:  133 x 104 matrix of allele counts
##    @loc.n.all: number of alleles per locus (range: 8-16)
##    @loc.fac: locus factor for the 104 columns of @tab
##    @all.names: list of allele names for each locus
##    @ploidy: ploidy of each individual  (range: 2-2)
##    @type:  codom
##    @call: read.genepop(file = "data/nancycats.gen", ncode = 3, quiet = TRUE)
## 
##  // Optional content
##    @pop: population of each individual (group size range: 10-25)

Geeky note: For more details on the content of the genind object, just try:

str(myData)

or for even more geeky details:

print.AsIs(myData)

As you will see, the genind object is a particular kind of list. Technically it is called a S4 object. Instead of elements accessible with $, S4 objects have slots accessible with @ (even if the programmer behind adegenet made it possible to use $ anyway). ___

Since manipulating such objects is a little complicated, you should use the accessors directly available:

nInd(myData)

## [1] 133

nLoc(myData)

## [1] 9

nPop(myData)

## [1] 8

locNames(myData)

## [1] "fca8"  "fca23" "fca43" "fca45" "fca77" "fca78" "fca90" "fca96" "fca37"

alleles(myData)

## $fca8
##  [1] "137" "141" "129" "133" "131" "135" "139" "147" "123" "149" "145" "143" "127" "119" "117" "121"
## 
## $fca23
##  [1] "130" "136" "146" "144" "138" "132" "142" "128" "140" "150" "148"
## 
## $fca43
## [1] "137" "145" "135" "133" "139" "149" "141" "147" "143"
## 
## $fca45
## [1] "128" "126" "130" "120" "118" "132" "122" "116" "134"
## 
## $fca77
##  [1] "152" "144" "150" "156" "146" "158" "148" "160" "154" "142" "162" "132"
## 
## $fca78
## [1] "142" "150" "140" "148" "146" "144" "138" "152"
## 
## $fca90
##  [1] "193" "199" "191" "185" "203" "197" "195" "201" "187" "205" "189" "181"
## 
## $fca96
##  [1] "091" "113" "101" "121" "107" "109" "117" "103" "115" "111" "119" "105"
## 
## $fca37
##  [1] "182" "208" "206" "192" "194" "214" "200" "220" "210" "218" "212" "186" "216" "204" "184"

indNames(myData)

##   [1] "N7"   "N141" "N142" "N143" "N144" "N145" "N146" "N147" "N148" "N149" "N151" "N153" "N154" "N155" "N156" "N157" "N158" "N159" "N160" "N161" "N162" "N163"
##  [23] "N463" "N35"  "N36"  "N37"  "N38"  "N39"  "N40"  "N41"  "N42"  "N44"  "N45"  "N46"  "N47"  "N48"  "N49"  "N50"  "N51"  "N52"  "N53"  "N54"  "N239" "N240"
##  [45] "N250" "N268" "N450" "N468" "N80"  "N81"  "N82"  "N83"  "N84"  "N85"  "N86"  "N87"  "N187" "N188" "N189" "N190" "N191" "N192" "N43"  "N92"  "N94"  "N95" 
##  [67] "N96"  "N98"  "N99"  "N100" "N93"  "N97"  "N125" "N126" "N127" "N128" "N129" "N130" "N131" "N132" "N133" "N246" "N247" "N271" "N298" "N299" "N300" "N301"
##  [89] "N302" "N303" "N304" "N310" "N193" "N194" "N195" "N196" "N197" "N198" "N199" "N200" "N201" "N202" "N203" "N206" "N207" "N209" "N210" "N211" "N212" "N204"
## [111] "N227" "N228" "N229" "N230" "N231" "N232" "N233" "N234" "N235" "N236" "N282" "N283" "N288" "N291" "N292" "N293" "N294" "N295" "N296" "N297" "N281" "N289"
## [133] "N290"

popNames(myData)

## [1] "N463" "N468" "N192" "N97"  "N310" "N212" "N236" "N290"

Some of the accessors can also be used to redefine some information (to handle with great care):

#let's give the pops some names:

#myPops <- c("Pop1","Pop2","Pop3","Pop4","Pop5","Pop6","Pop7", "Pop8")
myPops <- paste0("Pop", 1:nPop(myData))
popNames(myData) <- myPops
popNames(myData)

## [1] "Pop1" "Pop2" "Pop3" "Pop4" "Pop5" "Pop6" "Pop7" "Pop8"

pop(myData)

##   [1] Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop1 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2
##  [32] Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop2 Pop3 Pop3 Pop3 Pop3 Pop3 Pop3 Pop3 Pop3 Pop3 Pop3 Pop3 Pop3 Pop3 Pop3
##  [63] Pop4 Pop4 Pop4 Pop4 Pop4 Pop4 Pop4 Pop4 Pop4 Pop4 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop5 Pop6
##  [94] Pop6 Pop6 Pop6 Pop6 Pop6 Pop6 Pop6 Pop6 Pop6 Pop6 Pop6 Pop6 Pop6 Pop6 Pop6 Pop6 Pop7 Pop7 Pop7 Pop7 Pop7 Pop7 Pop7 Pop7 Pop7 Pop7 Pop7 Pop8 Pop8 Pop8 Pop8
## [125] Pop8 Pop8 Pop8 Pop8 Pop8 Pop8 Pop8 Pop8 Pop8
## Levels: Pop1 Pop2 Pop3 Pop4 Pop5 Pop6 Pop7 Pop8

The `loci` format

Not all R packages dealing with genetics/genomics use genind objects (it would be convenient if they did!). In particular, the package pegas which allows for the computation of many useful metrics relies on an alternative format called loci.

myData_loci <- genind2loci(myData) # or as.loci(myData)
myData_loci

## Allelic data frame: 133 individuals
##                     9 loci
##                     1 additional variable

The loci format is more simple than the geneind format (and so what you can do with it is more limited).

Geeky note: A loci object is simply a traditional data.frame (an S3 object) with an additional attribute called "locicol":

str(myData_loci)

## Classes 'loci' and 'data.frame': 133 obs. of  10 variables:
##  $ population: Factor w/ 8 levels "Pop1","Pop2",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ fca8      : Factor w/ 43 levels "123/123","123/143",..: 31 5 5 14 8 6 5 6 19 8 ...
##  $ fca23     : Factor w/ 34 levels "128/150","128/130",..: 4 4 3 4 15 19 7 22 18 4 ...
##  $ fca43     : Factor w/ 24 levels "133/133","135/135",..: 11 20 20 2 8 2 1 2 20 7 ...
##  $ fca45     : Factor w/ 27 levels "118/132","116/120",..: 18 17 19 19 12 19 11 9 11 9 ...
##  $ fca77     : Factor w/ 38 levels "142/142","144/144",..: 22 4 24 34 22 2 2 10 31 25 ...
##  $ fca78     : Factor w/ 23 levels "138/138","140/140",..: 10 2 6 6 5 6 2 4 18 10 ...
##  $ fca90     : Factor w/ 30 levels "181/185","185/185",..: 16 16 16 27 27 16 9 24 7 24 ...
##  $ fca96     : Factor w/ 31 levels "091/091","091/103",..: 1 7 1 1 1 1 1 7 1 9 ...
##  $ fca37     : Factor w/ 24 levels "182/182","182/192",..: 1 6 16 5 16 2 2 1 1 16 ...
##  - attr(*, "locicol")= int  2 3 4 5 6 7 8 9 10

head(data.frame(myData_loci), n = 10) # first 10 rows

##      population    fca8   fca23   fca43   fca45   fca77   fca78   fca90   fca96   fca37
## N7         Pop1 137/141 130/136 137/145 128/128 152/152 142/150 193/199 091/091 182/182
## N141       Pop1 129/133 130/136 135/145 126/128 144/150 140/140 193/199 091/113 182/208
## N142       Pop1 129/133 130/130 135/145 128/130 152/156 142/142 193/199 091/091 208/208
## N143       Pop1 133/133 130/136 135/135 128/130 156/156 142/142 199/199 091/091 182/206
## N144       Pop1 131/135 136/136 137/137 126/130 152/152 140/142 199/199 091/091 208/208
## N145       Pop1 129/135 136/146 135/135 128/130 144/144 142/142 193/199 091/091 182/192
## N146       Pop1 129/133 130/144 133/133 126/126 144/144 140/140 191/191 091/091 182/192
## N147       Pop1 129/135 138/138 135/135 120/126 146/158 140/148 191/199 091/113 182/182
## N148       Pop1 135/135 136/144 135/145 126/126 146/156 140/150 185/191 091/091 182/182
## N149       Pop1 131/135 130/136 135/137 120/126 152/158 142/150 191/199 101/121 208/208

attr(myData_loci, "locicol") # columns that contain loci

## [1]  2  3  4  5  6  7  8  9 10

Checking your data

Let’s check some basic things about the data in myData.

Missing data

An important consideration for analysis is the amount of missing data in our dataset. Several types of analyses don’t cope very well with missing data. In some cases this may lead to poor estimates, in other cases it may lead to samples or loci not being considered in the analysis.

We can use info_table() from poppr to find out about missing data, and also generate a convenient plot by including plot = TRUE.

info_table(myData, plot = TRUE, low = "white")

##           Locus
## Population  fca8 fca23 fca43 fca45 fca77 fca78 fca90 fca96 fca37  Mean
##      Pop1      .     .     .     .     .     .     .     .     .     .
##      Pop2      .     .     .     .     .     .     .     .     .     .
##      Pop3  0.357     .     .     .     .     .     .     .     . 0.040
##      Pop4      .     .     .     .     .     .     .     .     .     .
##      Pop5  0.150     .     . 0.400     .     .     . 0.050     . 0.067
##      Pop6  0.412     .     .     .     .     .     .     .     . 0.046
##      Pop7      .     .     .     .     .     .     .     .     .     .
##      Pop8      .     .     . 1.000     .     .     . 0.615     . 0.179
##      Total 0.113     .     . 0.158     .     .     . 0.068     . 0.038

Geeky note: it is tricky but you can modify anything in this plot even when info_table() as no option for it:

theplot <- ggplot_build(last_plot())
theplot$data[[4]]$size <- 3
theplot$data[[4]]$angle <- 45
plot(ggplot_gtable(theplot))

We can see that there is missing data at several loci, and that this is particularly bad for our population Pop8.

As a general rule of thumb, we try to keep missing data below 5% in order to minimize the impact on analyses.

How do you think we can achieve this?

Using genotype_curve() we can check how many loci we need in order to discriminate our genotypes. This poppr function randomly samples loci without replacement and counts the number of multilocus genotypes observed.

genotype_curve(myData)

How many loci do we *need* in order to discriminate our genotypes?

Geeky note: you can get the exact numbers:

gencurv <- genotype_curve(myData)

apply(gencurv, 2, range)

##       NumLoci
##         1   2   3   4   5   6   7   8
##   [1,] 23  74 111 126 128 129 130 130
##   [2,] 44 110 130 130 130 130 130 130

As missing data is concentrated in a few loci and populations, rather than being thinly spread throughout the dataset, we can remove the worst offenders: a locus (fca8) and a population (Pop8).

We can ‘drop’ a population from the genind object using the following code, if we know the number (ID) of the population in the list of populations.

myData[pop = -c(8)]

However, we would like to be able to remove populations by their name. For this we need a bit more code:

removePop <- c("Pop8")
myData <- myData[pop = !popNames(myData) %in% removePop]

Removing loci from a genind object works in a similar fashion. If you know the number (ID) of a locus, then you can use the short code myData[loc = -c(1)]. But again, it is better to use names.

removeLoc <- c("fca8")
myData <- myData[loc = !locNames(myData) %in% removeLoc]

Now let’s check myData and also run info_table() to see what’s changed.

myData

## /// GENIND OBJECT /////////
## 
##  // 120 individuals; 8 loci; 88 alleles; size: 69.1 Kb
## 
##  // Basic content
##    @tab:  120 x 88 matrix of allele counts
##    @loc.n.all: number of alleles per locus (range: 8-15)
##    @loc.fac: locus factor for the 88 columns of @tab
##    @all.names: list of allele names for each locus
##    @ploidy: ploidy of each individual  (range: 2-2)
##    @type:  codom
##    @call: .local(x = x, i = i, j = j, loc = ..1, drop = drop)
## 
##  // Optional content
##    @pop: population of each individual (group size range: 10-25)

info_table(myData, plot = TRUE, low = "white")

##           Locus
## Population fca23 fca43  fca45 fca77 fca78 fca90  fca96 fca37   Mean
##      Pop1      .     .      .     .     .     .      .     .      .
##      Pop2      .     .      .     .     .     .      .     .      .
##      Pop3      .     .      .     .     .     .      .     .      .
##      Pop4      .     .      .     .     .     .      .     .      .
##      Pop5      .     . 0.4000     .     .     . 0.0500     . 0.0563
##      Pop6      .     .      .     .     .     .      .     .      .
##      Pop7      .     .      .     .     .     .      .     .      .
##      Total     .     . 0.0667     .     .     . 0.0083     . 0.0094

We have significantly reduced the amount of missing data in myData.

How else could we have solved this?

How about our ability to discriminate genotypes?

genotype_curve(myData)

Have you noticed something?

MLG (MultiLocus Genotypes)

Depending on how we obtain our genetic samples, we may not know if we have genotyped the same individual more than once.

For example, if we have been genotyping samples collected in a non-invasive way (e.g. hair or faeces), then we don’t really have a way of knowing if we have obtained more than one sample per individual.

There is an R package called alleleMatch that can investigate this in detail. However, it is no longer maintained. Thankfully, poppr has some basic functionality to examine this. Using mlg() we can check for the number of unique multilocus genotypes in myData.

mlg(myData)

## #############################
## # Number of Individuals:  120 
## # Number of MLG:  117 
## #############################

## [1] 117

We have 120 samples, but only 117 unique multilocus genotypes. This suggests that 3 samples correspond to replicates from the same individual(s). (We introduce that in the data on purpose to show you how to deal with such issues.)

Note: the mlg() function did not tell us which samples are the same. For this we can use mlg.id(). But this gives us a very long list, because it also includes the genotypes that occur only once. Check using head():

head(mlg.id(myData))

## $`1`
## [1] "N247"
## 
## $`2`
## [1] "N95"
## 
## $`3`
## [1] "N133"
## 
## $`4`
## [1] "N84"
## 
## $`5`
## [1] "N190"
## 
## $`6`
## [1] "N80"

What we want to know is which are the genotypes that occur more than once?

We achieve this by looking for entries in the list with a length greater than one.

myMLG <- mlg.id(myData)
myMLG[lengths(myMLG) > 1]

## $`77`
## [1] "N268" "N468"
## 
## $`80`
## [1] "N250" "N450"
## 
## $`87`
## [1] "N163" "N463"

As we do not want to include replicates of samples in our data, we should now remove one of each replicated genotype. We will remove N450, N463, and N468.

myData <- myData[!indNames(myData) %in% c("N450","N463","N468")]
myData

## /// GENIND OBJECT /////////
## 
##  // 117 individuals; 8 loci; 88 alleles; size: 67.7 Kb
## 
##  // Basic content
##    @tab:  117 x 88 matrix of allele counts
##    @loc.n.all: number of alleles per locus (range: 8-15)
##    @loc.fac: locus factor for the 88 columns of @tab
##    @all.names: list of allele names for each locus
##    @ploidy: ploidy of each individual  (range: 2-2)
##    @type:  codom
##    @call: .local(x = x, i = i, j = j, drop = drop)
## 
##  // Optional content
##    @pop: population of each individual (group size range: 10-23)

Now we have 117 unique genotypes in myData.

If this were data from non-invasively collected samples, we may expect there to be more duplicate genotypes that we have not detected yet. Why?

Geeky note:

If you have to remove a lot of samples, you can do it easily by replacing the previous call by:

samplesToKeep <- unlist(lapply(myMLG, function(x) x[1]))
myData <- myData[indNames(myData) %in% samplesToKeep]

Ploidy

info_table() can also be used to find out about the ploidy of your data. As the results are presented for all samples, here just the first couple lines of output (achieved by using head())

head(info_table(myData, type = "ploidy"))

##        Loci
## Samples fca23 fca43 fca45 fca77 fca78 fca90 fca96 fca37
##    N7       2     2     2     2     2     2     2     2
##    N141     2     2     2     2     2     2     2     2
##    N142     2     2     2     2     2     2     2     2
##    N143     2     2     2     2     2     2     2     2
##    N144     2     2     2     2     2     2     2     2
##    N145     2     2     2     2     2     2     2     2

table(info_table(myData, type = "ploidy"), useNA = "always") ## counts different values

## 
##    2 <NA> 
##  927    9

nLoc(myData) * nInd(myData) ## total

## [1] 936

How many alleles expect in a tetraploid?
Could this number be 1 for a diploid?

Are loci informative

Another important consideration is if our loci are variable and informative. Otherwise we’re wasting time and space on keeping them! The function informloci() can be used to detect uninformative loci and remove them.

informloci(myData)

## cutoff value: 1.70940170940171 % ( 2 samples ).

## MAF         : 0.01

## 
## All sites polymorphic

## /// GENIND OBJECT /////////
## 
##  // 117 individuals; 8 loci; 88 alleles; size: 67.8 Kb
## 
##  // Basic content
##    @tab:  117 x 88 matrix of allele counts
##    @loc.n.all: number of alleles per locus (range: 8-15)
##    @loc.fac: locus factor for the 88 columns of @tab
##    @all.names: list of allele names for each locus
##    @ploidy: ploidy of each individual  (range: 2-2)
##    @type:  codom
##    @call: .local(x = x, i = i, j = j, loc = ..1, drop = drop)
## 
##  // Optional content
##    @pop: population of each individual (group size range: 10-23)

In our case, all loci are informative and they are kept. If we have uninformative loci, and want these to be removed, we need to run the following.

myData <- informloci(myData)

Genetic variation

A very simple exploration

You can easily explore genetic differences using our home made little function:

#all_pairwise <- pairwise_similarity(myData)
#hist(all_pairwise$ratio)
lapply(seppop(myData), function(x) head(pairwise_similarity(x)))

## $Pop1
##    id1  id2 distinct common total     ratio
## 1 N141   N7        7     12    19 0.6315789
## 2 N142   N7        4     12    16 0.7500000
## 3 N142 N141        4     15    19 0.7894737
## 4 N143   N7        4     12    16 0.7500000
## 5 N143 N141        4     15    19 0.7894737
## 6 N143 N142        3     12    15 0.8000000
## 
## $Pop2
##   id1 id2 distinct common total     ratio
## 1 N36 N35        5     13    18 0.7222222
## 2 N37 N35        6     13    19 0.6842105
## 3 N37 N36        4     14    18 0.7777778
## 4 N38 N35       11     13    24 0.5416667
## 5 N38 N36       10     14    24 0.5833333
## 6 N38 N37        8     11    19 0.5789474
## 
## $Pop3
##   id1 id2 distinct common total     ratio
## 1 N81 N80        8     13    21 0.6190476
## 2 N82 N80        7     13    20 0.6500000
## 3 N82 N81        3     14    17 0.8235294
## 4 N83 N80        9     13    22 0.5909091
## 5 N83 N81        8     14    22 0.6363636
## 6 N83 N82        7     13    20 0.6500000
## 
## $Pop4
##   id1 id2 distinct common total     ratio
## 1 N92 N43       13     12    25 0.4800000
## 2 N94 N43        9     12    21 0.5714286
## 3 N94 N92        9     16    25 0.6400000
## 4 N95 N43        6     12    18 0.6666667
## 5 N95 N92        8     16    24 0.6666667
## 6 N95 N94        9     12    21 0.5714286
## 
## $Pop5
##    id1  id2 distinct common total     ratio
## 1 N126 N125        9     14    23 0.6086957
## 2 N127 N125        7     14    21 0.6666667
## 3 N127 N126        6     13    19 0.6842105
## 4 N128 N125        5     14    19 0.7368421
## 5 N128 N126        6     13    19 0.6842105
## 6 N128 N127        5     15    20 0.7500000
## 
## $Pop6
##    id1  id2 distinct common total     ratio
## 1 N194 N193        5     14    19 0.7368421
## 2 N195 N193        6     14    20 0.7000000
## 3 N195 N194        6     12    18 0.6666667
## 4 N196 N193        5     14    19 0.7368421
## 5 N196 N194        4     12    16 0.7500000
## 6 N196 N195        7     12    19 0.6315789
## 
## $Pop7
##    id1  id2 distinct common total     ratio
## 1 N227 N204       10     14    24 0.5833333
## 2 N228 N204       10     14    24 0.5833333
## 3 N228 N227        4     14    18 0.7777778
## 4 N229 N204       11     14    25 0.5600000
## 5 N229 N227        6     14    20 0.7000000
## 6 N229 N228        6     15    21 0.7142857

par(mfrow = c(3, 3))
lapply(seppop(myData), function(x) hist(pairwise_similarity(x)$ratio, xlim = c(0, 1)))

## $Pop1
## $breaks
##  [1] 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
## 
## $counts
##  [1]  9 20 57 55 41 28 12  5  3  1
## 
## $density
##  [1] 0.77922078 1.73160173 4.93506494 4.76190476 3.54978355 2.42424242 1.03896104 0.43290043 0.25974026 0.08658009
## 
## $mids
##  [1] 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925
## 
## $xname
## [1] "pairwise_similarity(x)$ratio"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## $Pop2
## $breaks
##  [1] 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90
## 
## $counts
## [1] 14 30 65 75 49 13  5  0  2
## 
## $density
## [1] 1.1067194 2.3715415 5.1383399 5.9288538 3.8735178 1.0276680 0.3952569 0.0000000 0.1581028
## 
## $mids
## [1] 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875
## 
## $xname
## [1] "pairwise_similarity(x)$ratio"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## $Pop3
## $breaks
## [1] 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
## 
## $counts
## [1]  9 24 24 14 11  6  2  1
## 
## $density
## [1] 1.9780220 5.2747253 5.2747253 3.0769231 2.4175824 1.3186813 0.4395604 0.2197802
## 
## $mids
## [1] 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925
## 
## $xname
## [1] "pairwise_similarity(x)$ratio"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## $Pop4
## $breaks
##  [1] 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90
## 
## $counts
## [1]  3  5 12  9 12  3  0  0  1
## 
## $density
## [1] 1.3333333 2.2222222 5.3333333 4.0000000 5.3333333 1.3333333 0.0000000 0.0000000 0.4444444
## 
## $mids
## [1] 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875
## 
## $xname
## [1] "pairwise_similarity(x)$ratio"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## $Pop5
## $breaks
##  [1] 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
## 
## $counts
##  [1]  2 12 40 44 46 27 11  4  2  1  1
## 
## $density
##  [1] 0.2105263 1.2631579 4.2105263 4.6315789 4.8421053 2.8421053 1.1578947 0.4210526 0.2105263 0.1052632 0.1052632
## 
## $mids
##  [1] 0.425 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925
## 
## $xname
## [1] "pairwise_similarity(x)$ratio"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## $Pop6
## $breaks
## [1] 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85
## 
## $counts
## [1] 13 24 42 20 22 10  4  1
## 
## $density
## [1] 1.9117647 3.5294118 6.1764706 2.9411765 3.2352941 1.4705882 0.5882353 0.1470588
## 
## $mids
## [1] 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825
## 
## $xname
## [1] "pairwise_similarity(x)$ratio"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## $Pop7
## $breaks
## [1] 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90
## 
## $counts
## [1]  1  8 15 16  9  4  1  1
## 
## $density
## [1] 0.3636364 2.9090909 5.4545455 5.8181818 3.2727273 1.4545455 0.3636364 0.3636364
## 
## $mids
## [1] 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875
## 
## $xname
## [1] "pairwise_similarity(x)$ratio"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"

# lapply(seppop(myData), pairwise_similarity) ## for all
par(mfrow = c(1, 1))

Allele vs genotype frequencies

Consider the following two populations with different genotype frequencies:

             AA    Aa    aa
      Pop1   50     0    50
      Pop2   25    50    25

Do the genotype frequencies of the two populations differ?

Do the allele frequencies of the two populations differ?

Do you remember …

?

... what Hardy-Weinberg equilibrium (HWE) is?

Assumptions

Genotype frequencies are the same in males and females.
Genotypes mate at random with respect to their genotype at this particular locus (panmixia).
Meiosis is fair.
There is no new genetic material (no mutation).
There is no gene flow (no migration).
The population is of infinite size (no drift).
All matings produce the same average number of offspring (no selection on fecundity).
There are no differences among genotypes in the probability of survival (no selection on survival).
Generations do not overlap (no selection on reproductive rate).

Freq. in next generation

Using frequency of alleles in the current generation …

$p_t = f(A)$

$q_t = f(a)$

… the frequencies in the next generation can be calculated:

$f(AA) = p_t^{2}$

$f(Aa) = 2p_tq_t$

$f(aa) = q_t^{2}$

$p_{t+1} = f(AA) + \dfrac{f(Aa)}{2} = p_t^2 + \dfrac{2p_tq_t}{2} = p_t^2 + p_tq_t = p_t^2 + p_t(1-p_t) = p_t^2 + p_t - p_t^2 = p_t$

$q_{t+1} = f(aa) + \dfrac{f(Aa)}{2} = q_t^2 + \dfrac{2p_tq_t}{2} = q_t^2 + p_tq_t = q_t^2 + (1-q_t)q_t = q_t^2 + q_t - q_t^2 = q_t$

Under HWE, the allelic frequencies remain the same over time!

Why do we care?

Evolution is a change in allele frequencies in a population over time

When a locus in a population is in HWE, it is not evolving: allele frequencies will stay the same across generations.

If HWE assumptions are not met, evolution can happen (allele frequencies may change).

Mutation, non-random mating, gene flow, genetic drift (caused by finite population size), and natural selection violate HWE assumptions and are thus “mechanisms” by which evolution may proceed.

The HWE is thus a the null model of micro-evolution.

So is it good/bad for a locus to be in HWE?

“Neutral” evolution

Very broadly speaking, there are two types of population genetics analyses you can do:

those that assume loci are in HWE, and
those that do not.

It is thus important to realise if your data reject the HWE assumptions!

Testing the HWE

With hw.test() from pegas we can test for HWE across our populations:

hw.test(myData, B = 0) ## for exact test instead of asymptotic, use large value of B

##           chi^2  df  Pr(chi^2 >)
## fca23 153.14590  55 3.417067e-11
## fca43 169.84216  36 0.000000e+00
## fca45  46.60491  36 1.109412e-01
## fca77 194.25536  66 1.454392e-14
## fca78 248.27891  28 0.000000e+00
## fca90 256.37179  66 0.000000e+00
## fca96 339.76874  66 0.000000e+00
## fca37 298.50220 105 0.000000e+00

Here we have conducted a $\chi^{2}$-test based on observed and expected genotype frequencies calculated from the allele frequencies.

?

Would we expect loci to be in HWE if we consider all populations together as we have done?

Wahlund

The Wahlund effect refers to the reduction in the number of heterozygotes due to population structure.

Consider two populations:

             AA    Aa    aa
      Pop1   50     0     0
      Pop2    0     0    50

Here, the two populations are in HWE.

However, if we treat them as a single population, there are no heterozygotes, and this merged-population is not in HWE.

Let’s check by population. We will split myData by population using the seppop function. This creates a list of genind objects, with every entry in the list consisting of a population.

To apply the HWE test with hw.test() to every item in the list (ie every population), we use the function lapply().

lapply(seppop(myData), hw.test)

## $Pop1
##          chi^2 df  Pr(chi^2 >) Pr.exact
## fca23 23.22222 10 9.955382e-03    0.255
## fca43 27.20034  6 1.328128e-04    0.001
## fca45 12.21716 15 6.625229e-01    0.458
## fca77 86.16667 45 2.152672e-04    0.000
## fca78 11.50468  6 7.397613e-02    0.008
## fca90 56.75227 15 9.039990e-07    0.000
## fca96 92.07157 15 4.066747e-13    0.000
## fca37 13.17284 15 5.889498e-01    0.211
## 
## $Pop2
##          chi^2 df  Pr(chi^2 >) Pr.exact
## fca23 30.85879 28 0.3233864330    0.091
## fca43 10.77193 15 0.7685881191    0.794
## fca45 26.58116 21 0.1851504896    0.167
## fca77 35.34014 21 0.0259007804    0.014
## fca78 21.05849 15 0.1349710053    0.025
## fca90 61.74736 28 0.0002424509    0.011
## fca96 14.23444 21 0.8593197525    0.271
## fca37 25.96444 28 0.5749990390    0.116
## 
## $Pop3
##           chi^2 df Pr(chi^2 >) Pr.exact
## fca23 29.793651 15 0.012687711    0.007
## fca43  3.718750 10 0.959144052    0.750
## fca45  5.685319  6 0.459347253    0.376
## fca77  5.780408  6 0.448233208    0.175
## fca78 38.502857 15 0.000760033    0.003
## fca90  7.767857  6 0.255608831    0.186
## fca96  3.982222  6 0.679082361    0.425
## fca37  4.709877  3 0.194316502    0.101
## 
## $Pop4
##           chi^2 df Pr(chi^2 >) Pr.exact
## fca23 10.513889 10 0.396621136    0.551
## fca43 30.000000 21 0.091988007    0.199
## fca45 11.521794 10 0.318334361    0.402
## fca77 39.400000 21 0.008789371    0.001
## fca78 34.444444 21 0.032459406    0.001
## fca90  7.155556  6 0.306700905    0.369
## fca96 12.233560 15 0.661271220    0.529
## fca37 33.511111 21 0.040848059    0.010
## 
## $Pop5
##          chi^2 df Pr(chi^2 >) Pr.exact
## fca23 58.52901 45 0.084923238    0.186
## fca43 26.97092 21 0.171817797    0.016
## fca45 28.01333 28 0.463741139    0.017
## fca77 50.64042 36 0.053561033    0.000
## fca78 15.01731 10 0.131431929    0.023
## fca90 60.22659 36 0.006894609    0.094
## fca96 39.55940 28 0.072335908    0.278
## fca37 30.97222 21 0.074121484    0.031
## 
## $Pop6
##           chi^2 df  Pr(chi^2 >) Pr.exact
## fca23 49.799592 15 1.298351e-05    0.001
## fca43 31.801084 21 6.132040e-02    0.042
## fca45  7.930211 10 6.356538e-01    0.604
## fca77 27.672222 15 2.371323e-02    0.008
## fca78 28.097222 15 2.096833e-02    0.096
## fca90 30.337654 21 8.540995e-02    0.006
## fca96 58.085432 36 1.128794e-02    0.017
## fca37  0.489298  3 9.212362e-01    0.854
## 
## $Pop7
##           chi^2 df Pr(chi^2 >) Pr.exact
## fca23  5.228395  6 0.514871229    0.595
## fca43  4.820988  6 0.566969462    0.663
## fca45  6.302083 10 0.789277136    0.893
## fca77  7.260000 10 0.700692473    0.721
## fca78 18.740741  6 0.004624641    0.019
## fca90 18.122500 15 0.256280157    0.279
## fca96 26.510000 15 0.032991782    0.055
## fca37 12.222222  6 0.057190992    0.254

unlist(lapply(seppop(myData), nInd)) ## careful: pop size differ

## Pop1 Pop2 Pop3 Pop4 Pop5 Pop6 Pop7 
##   22   23   14   10   20   17   11

# lapply(seppop(myData), function(x) hw.test(x)) ## SAME THING

Some loci, in some populations, are still not in HWE.

Why may this be?

Linkage Disequilibrium

Another important consideration for analysing genetic data is the independence of loci.

Imagine two loci that are physically very close on the same chromosome. Will alleles at these loci be segregating independently?

Do you know how this is measured?

For this reason, and many others (e.g. Fisher’s runaway), it is possible that alleles at one locus may be associated with alleles at another locus. In other words, we may observe non-random association of alleles at different loci. This is called linkage disequilibrium.

Why is this important? If two loci are not independent this will bias results.

We can look at this with the ia() function of poppr. This calculates an index of association over all loci in the genind object.

ia(myData, sample = 999)

##          Ia        p.Ia       rbarD        p.rD 
## 0.032231701 0.371000000 0.004646249 0.371000000

Here we can see the distribution of expected association between alleles at different loci when there is no linkage (dark grey barplot), and the estimate for association among alleles in our total dataset (ie all loci and all pops at the same time).

It appears we do not have a significant association across all populations; i.e., no LD.

What if we look at the same test per population? To answer this question, we use seppop() and lapply() as before. But now we include ia() and for the argument sample, which defines the number of permutations used to draw the distribution of the association index under the null hypothesis, we want a large number, e.g., 999:

lapply(seppop(myData), ia, sample = 999)

## $Pop1
##         Ia       p.Ia      rbarD       p.rD 
## 0.24839705 0.02600000 0.03597463 0.02500000 
## 
## $Pop2
##         Ia       p.Ia      rbarD       p.rD 
## 0.34478904 0.00500000 0.04940941 0.00500000 
## 
## $Pop3
##         Ia       p.Ia      rbarD       p.rD 
## 0.22960049 0.08600000 0.03294858 0.08800000 
## 
## $Pop4
##         Ia       p.Ia      rbarD       p.rD 
## 0.24210526 0.16000000 0.03494786 0.16100000 
## 
## $Pop5
##         Ia       p.Ia      rbarD       p.rD 
## 0.09058140 0.34800000 0.01313577 0.34700000 
## 
## $Pop6
##         Ia       p.Ia      rbarD       p.rD 
## 0.25275944 0.04100000 0.03638827 0.04100000 
## 
## $Pop7
##         Ia       p.Ia      rbarD       p.rD 
## 0.54708418 0.00700000 0.07877947 0.00700000

# lapply(seppop(myData), function(x) ia(x, sample = 999)) ## SAME THING

In some populations we observe significant LD.

Is this important?

If we want to know which two loci are in LD, we can use the pair.ia() function. Let’s consider Pop7 using seppop(myData)[["Pop7"]].

pair.ia(seppop(myData)[["Pop7"]])

##                  Ia   rbarD
## fca23:fca43  0.4763  0.4800
## fca23:fca45  0.2649  0.2652
## fca23:fca77  0.4578  0.4578
## fca23:fca78 -0.1304 -0.1318
## fca23:fca90  0.1356  0.1357
## fca23:fca96  0.0305  0.0306
## fca23:fca37 -0.0434 -0.0437
## fca43:fca45  0.2775  0.2785
## fca43:fca77  0.2721  0.2745
## fca43:fca78  0.1330  0.1378
## fca43:fca90  0.2165  0.2174
## fca43:fca96  0.1113  0.1139
## fca43:fca37 -0.1151 -0.1183
## fca45:fca77  0.0020  0.0020
## fca45:fca78  0.0458  0.0466
## fca45:fca90  0.1371  0.1371
## fca45:fca96  0.0361  0.0364
## fca45:fca37  0.1287  0.1302
## fca77:fca78 -0.2098 -0.2116
## fca77:fca90  0.1019  0.1020
## fca77:fca96  0.0302  0.0303
## fca77:fca37  0.2262  0.2273
## fca78:fca90  0.1418  0.1440
## fca78:fca96  0.2988  0.2992
## fca78:fca37 -0.1809 -0.1810
## fca90:fca96 -0.0090 -0.0091
## fca90:fca37 -0.1912 -0.1932
## fca96:fca37 -0.1527 -0.1527

We can see that LD is not the same for all pairs of loci. Or can we? (Dany cannot!)

Let us change the plot so that it is easier for colour-blind people to see. For this we add some additional arguments for colour: low = "black" and high = "green".

pair.ia(myData[pop = "Pop7"], low = "black", high = "green")

##                  Ia   rbarD
## fca23:fca43  0.4763  0.4800
## fca23:fca45  0.2649  0.2652
## fca23:fca77  0.4578  0.4578
## fca23:fca78 -0.1304 -0.1318
## fca23:fca90  0.1356  0.1357
## fca23:fca96  0.0305  0.0306
## fca23:fca37 -0.0434 -0.0437
## fca43:fca45  0.2775  0.2785
## fca43:fca77  0.2721  0.2745
## fca43:fca78  0.1330  0.1378
## fca43:fca90  0.2165  0.2174
## fca43:fca96  0.1113  0.1139
## fca43:fca37 -0.1151 -0.1183
## fca45:fca77  0.0020  0.0020
## fca45:fca78  0.0458  0.0466
## fca45:fca90  0.1371  0.1371
## fca45:fca96  0.0361  0.0364
## fca45:fca37  0.1287  0.1302
## fca77:fca78 -0.2098 -0.2116
## fca77:fca90  0.1019  0.1020
## fca77:fca96  0.0302  0.0303
## fca77:fca37  0.2262  0.2273
## fca78:fca90  0.1418  0.1440
## fca78:fca96  0.2988  0.2992
## fca78:fca37 -0.1809 -0.1810
## fca90:fca96 -0.0090 -0.0091
## fca90:fca37 -0.1912 -0.1932
## fca96:fca37 -0.1527 -0.1527

Much better ;)

lapply(seppop(myData), pair.ia, low = "black", high = "green")

## $Pop1
##                  Ia   rbarD
## fca23:fca43  0.0337  0.0338
## fca23:fca45  0.0038  0.0039
## fca23:fca77 -0.0373 -0.0376
## fca23:fca78  0.0306  0.0308
## fca23:fca90  0.0133  0.0134
## fca23:fca96 -0.0257 -0.0264
## fca23:fca37  0.1288  0.1288
## fca43:fca45  0.1943  0.1966
## fca43:fca77  0.1966  0.2021
## fca43:fca78  0.0120  0.0120
## fca43:fca90 -0.0328 -0.0329
## fca43:fca96  0.0203  0.0205
## fca43:fca37  0.1115  0.1122
## fca45:fca77 -0.0435 -0.0436
## fca45:fca78  0.0159  0.0161
## fca45:fca90  0.1829  0.1866
## fca45:fca96  0.0047  0.0049
## fca45:fca37  0.0033  0.0033
## fca77:fca78  0.0939  0.0970
## fca77:fca90 -0.1271 -0.1321
## fca77:fca96  0.0733  0.0780
## fca77:fca37  0.1483  0.1493
## fca78:fca90 -0.0163 -0.0163
## fca78:fca96  0.0518  0.0520
## fca78:fca37  0.0110  0.0111
## fca90:fca96  0.0233  0.0233
## fca90:fca37 -0.0201 -0.0204
## fca96:fca37 -0.0019 -0.0020
## 
## $Pop2
##                  Ia   rbarD
## fca23:fca43 -0.1198 -0.1198
## fca23:fca45  0.1944  0.1946
## fca23:fca77 -0.0536 -0.0536
## fca23:fca78 -0.0027 -0.0027
## fca23:fca90  0.0944  0.0944
## fca23:fca96  0.0218  0.0218
## fca23:fca37 -0.0684 -0.0691
## fca43:fca45 -0.0920 -0.0923
## fca43:fca77  0.1240  0.1240
## fca43:fca78  0.1349  0.1351
## fca43:fca90 -0.0166 -0.0167
## fca43:fca96  0.0423  0.0423
## fca43:fca37  0.4216  0.4279
## fca45:fca77 -0.0523 -0.0523
## fca45:fca78 -0.1235 -0.1236
## fca45:fca90  0.1536  0.1538
## fca45:fca96  0.2229  0.2230
## fca45:fca37  0.1505  0.1513
## fca77:fca78  0.1277  0.1277
## fca77:fca90 -0.0306 -0.0306
## fca77:fca96  0.0420  0.0420
## fca77:fca37  0.0110  0.0111
## fca78:fca90 -0.0098 -0.0098
## fca78:fca96 -0.0907 -0.0907
## fca78:fca37 -0.0491 -0.0494
## fca90:fca96  0.1768  0.1768
## fca90:fca37  0.0809  0.0819
## fca96:fca37  0.0537  0.0541
## 
## $Pop3
##                   Ia    rbarD
## fca23:fca43 -8.7e-02 -8.7e-02
## fca23:fca45 -5.6e-02 -5.6e-02
## fca23:fca77  3.6e-02  3.6e-02
## fca23:fca78 -1.0e-01 -1.0e-01
## fca23:fca90  1.1e-01  1.1e-01
## fca23:fca96  2.4e-01  2.4e-01
## fca23:fca37  3.1e-02  3.2e-02
## fca43:fca45  6.3e-02  6.3e-02
## fca43:fca77  6.0e-02  6.0e-02
## fca43:fca78  2.7e-01  2.7e-01
## fca43:fca90  1.5e-01  1.5e-01
## fca43:fca96 -1.5e-01 -1.5e-01
## fca43:fca37  6.7e-02  6.8e-02
## fca45:fca77 -1.1e-01 -1.1e-01
## fca45:fca78  5.3e-02  5.3e-02
## fca45:fca90  2.2e-01  2.2e-01
## fca45:fca96 -1.7e-01 -1.7e-01
## fca45:fca37  3.6e-01  3.6e-01
## fca77:fca78 -6.9e-02 -6.9e-02
## fca77:fca90 -7.5e-02 -7.5e-02
## fca77:fca96  7.4e-02  7.4e-02
## fca77:fca37 -1.1e-01 -1.1e-01
## fca78:fca90 -3.3e-02 -3.4e-02
## fca78:fca96  5.0e-02  5.0e-02
## fca78:fca37  2.4e-02  2.4e-02
## fca90:fca96  2.2e-16  2.8e-16
## fca90:fca37  3.4e-02  3.5e-02
## fca96:fca37  5.5e-02  5.5e-02
## 
## $Pop4
##                  Ia   rbarD
## fca23:fca43  0.1370  0.1371
## fca23:fca45  0.4550  0.4552
## fca23:fca77 -0.2619 -0.2650
## fca23:fca78  0.0303  0.0307
## fca23:fca90 -0.0847 -0.0847
## fca23:fca96  0.0264  0.0265
## fca23:fca37  0.0043  0.0043
## fca43:fca45 -0.0060 -0.0060
## fca43:fca77  0.1261  0.1284
## fca43:fca78  0.0145  0.0145
## fca43:fca90 -0.1102 -0.1103
## fca43:fca96  0.0690  0.0691
## fca43:fca37 -0.1377 -0.1393
## fca45:fca77 -0.2335 -0.2351
## fca45:fca78  0.0106  0.0108
## fca45:fca90 -0.1227 -0.1227
## fca45:fca96  0.1650  0.1665
## fca45:fca37  0.2585  0.2593
## fca77:fca78 -0.0755 -0.0791
## fca77:fca90  0.0811  0.0818
## fca77:fca96 -0.0165 -0.0171
## fca77:fca37 -0.1349 -0.1350
## fca78:fca90 -0.0731 -0.0741
## fca78:fca96  0.0155  0.0155
## fca78:fca37 -0.0627 -0.0649
## fca90:fca96  0.1749  0.1761
## fca90:fca37  0.3110  0.3124
## fca96:fca37  0.4809  0.4921
## 
## $Pop5
##                  Ia   rbarD
## fca23:fca43  0.0727  0.0732
## fca23:fca45 -0.1368 -0.1440
## fca23:fca77 -0.0169 -0.0169
## fca23:fca78  0.0582  0.0589
## fca23:fca90  0.1552  0.1553
## fca23:fca96  0.0853  0.0866
## fca23:fca37  0.2506  0.2564
## fca43:fca45  0.0403  0.0412
## fca43:fca77  0.0763  0.0769
## fca43:fca78  0.0261  0.0261
## fca43:fca90  0.1118  0.1128
## fca43:fca96  0.0041  0.0041
## fca43:fca37 -0.0122 -0.0122
## fca45:fca77  0.0286  0.0302
## fca45:fca78 -0.3668 -0.3727
## fca45:fca90 -0.1735 -0.1837
## fca45:fca96 -0.2480 -0.2508
## fca45:fca37 -0.2701 -0.2717
## fca77:fca78  0.1718  0.1736
## fca77:fca90  0.1279  0.1279
## fca77:fca96  0.1909  0.1938
## fca77:fca37 -0.1294 -0.1324
## fca78:fca90  0.1912  0.1937
## fca78:fca96  0.2730  0.2731
## fca78:fca37  0.1758  0.1762
## fca90:fca96 -0.1090 -0.1109
## fca90:fca37  0.0945  0.0970
## fca96:fca37  0.0825  0.0826
## 
## $Pop6
##                 Ia  rbarD
## fca23:fca43  0.128  0.129
## fca23:fca45  0.049  0.050
## fca23:fca77  0.333  0.333
## fca23:fca78  0.080  0.080
## fca23:fca90  0.021  0.021
## fca23:fca96 -0.077 -0.078
## fca23:fca37  0.132  0.133
## fca43:fca45 -0.145 -0.145
## fca43:fca77  0.222  0.227
## fca43:fca78  0.186  0.187
## fca43:fca90 -0.038 -0.038
## fca43:fca96 -0.015 -0.015
## fca43:fca37 -0.124 -0.124
## fca45:fca77 -0.140 -0.143
## fca45:fca78 -0.102 -0.103
## fca45:fca90  0.153  0.153
## fca45:fca96 -0.117 -0.117
## fca45:fca37  0.048  0.048
## fca77:fca78 -0.152 -0.153
## fca77:fca90  0.015  0.015
## fca77:fca96 -0.032 -0.033
## fca77:fca37  0.131  0.133
## fca78:fca90  0.157  0.157
## fca78:fca96  0.120  0.121
## fca78:fca37 -0.069 -0.069
## fca90:fca96 -0.052 -0.053
## fca90:fca37  0.179  0.179
## fca96:fca37 -0.023 -0.023
## 
## $Pop7
##                  Ia   rbarD
## fca23:fca43  0.4763  0.4800
## fca23:fca45  0.2649  0.2652
## fca23:fca77  0.4578  0.4578
## fca23:fca78 -0.1304 -0.1318
## fca23:fca90  0.1356  0.1357
## fca23:fca96  0.0305  0.0306
## fca23:fca37 -0.0434 -0.0437
## fca43:fca45  0.2775  0.2785
## fca43:fca77  0.2721  0.2745
## fca43:fca78  0.1330  0.1378
## fca43:fca90  0.2165  0.2174
## fca43:fca96  0.1113  0.1139
## fca43:fca37 -0.1151 -0.1183
## fca45:fca77  0.0020  0.0020
## fca45:fca78  0.0458  0.0466
## fca45:fca90  0.1371  0.1371
## fca45:fca96  0.0361  0.0364
## fca45:fca37  0.1287  0.1302
## fca77:fca78 -0.2098 -0.2116
## fca77:fca90  0.1019  0.1020
## fca77:fca96  0.0302  0.0303
## fca77:fca37  0.2262  0.2273
## fca78:fca90  0.1418  0.1440
## fca78:fca96  0.2988  0.2992
## fca78:fca37 -0.1809 -0.1810
## fca90:fca96 -0.0090 -0.0091
## fca90:fca37 -0.1912 -0.1932
## fca96:fca37 -0.1527 -0.1527

Descriptive statistics

There are some typical descriptive statistics you will find in most population genetics papers. These summary statistics give us an overview of some important features of populations under study.

We will focus on the basic ones describing variation within populations and describing variation between populations.

Note: many of these can be calculated over all populations as well as per population or between pairs of populations. This offers information at different scales.

Heterozygosity

If we are interested in genetic variation in natural populations we often consider heterozygosity.

High heterozygosity means a lot of genetic variability.

Low heterozygosity means little genetic variability.

Let’s consider heterozygosity in a simple system with two alleles at a locus: A and a. Let’s also assume that this population is in HWE.

Then:

$p = f(A)$

$q = f(a)$

So under HWE, we obtain the following mating table:

        A        a
  A     p^2      pq
  a     qp       q^2

So $pq + qp = 2pq$ gives the frequency of heterozygote genotypes.

In this two-allele system, heterozygosity is highest at $p = 0.5$. Let’s visualize:

freqP <- seq(from = 0, to = 1, by = 0.01)
freqQ <- 1 - freqP

plot(2*freqP*freqQ ~ freqQ, type = "l",
     ylab = "frequency of heterozygote genotypes",
     xlab = "frequency of allele 'a'")
text(0, 0, "AA")
text(1, 0, "aa")
arrows(0.45, 0, 0.05, 0)
text(0.5, 0, "Aa")
arrows(0.55, 0, 0.95, 0)

As you can imagine, this becomes more complex when there are more alleles per locus!

Questions?

Now we will calculate the observed heterozygosity in our populations, as well as the expected heterozygosity if the populations are in HWE.

For this we use the summary() function from adegenet. We can extract the observed heterozygosity using $Hobs and the expected heterozygosity using $Hexp.

To make life convenient, you can extract the values for every locus, place them in a data.frame and compute the difference:

heteroz <- data.frame(Hexp = summary(myData)$Hexp, Hobs = summary(myData)$Hobs)
heteroz$diff <- heteroz$Hexp - heteroz$Hobs
heteroz

##            Hexp      Hobs       diff
## fca23 0.8044415 0.6666667 0.13777486
## fca43 0.7754767 0.6581197 0.11735700
## fca45 0.7687484 0.7155963 0.05315209
## fca77 0.8734020 0.6153846 0.25801739
## fca78 0.7748922 0.5982906 0.17660165
## fca90 0.8141939 0.6837607 0.13043319
## fca96 0.7565770 0.5948276 0.16174941
## fca37 0.6576813 0.4957265 0.16195485

We can also visualize how observed heterozygosity is different from expected heterozygosity by subtracting one from the other.

If you remember from before, we saw that loci were not in HWE over all populations.

barplot(heteroz$diff, names.arg = rownames(heteroz),
        main = "Heterozygosity: expected-observed",
        xlab = "locus", ylab = "Hexp - Hobs", font.lab = 2)

Or we can use another representation, using ggplot2 for a change:

heteroz$loci <- rownames(heteroz) ## ggplot needs names stored as a column

ggplot(heteroz, aes(y = Hexp, x = Hobs)) +
  geom_segment(aes(yend = Hobs, xend = Hobs), linetype = "dashed") +
  geom_label(aes(label = loci)) +
  geom_abline(slope = 1) + 
  scale_x_continuous(limits = range(c(heteroz$Hobs, heteroz$Hexp))) +
  scale_y_continuous(limits = range(c(heteroz$Hobs, heteroz$Hexp))) +
  labs(title = "Heterozygosity: expected vs observed") +
  xlab(expression(bold("Observed heterozygosity"))) +
  ylab(expression(bold("Expected heterozygosity"))) +
  theme_classic()

Now let’s consider observed and expected heterozygosity by population. Here, we will focus on the mean across loci.

adegenet conveniently provides us with a funtion to calculate this for expected heterozygosity with Hs().

Hs(myData)

##      Pop1      Pop2      Pop3      Pop4      Pop5      Pop6      Pop7 
## 0.6747159 0.7258979 0.6246811 0.7206250 0.7556682 0.7480536 0.6854339

To obtain observed heterozygosity there is no short cut (yet), so we need to use lapply() and seppop() again. Here we will calculate the mean of $Hobs per population in myData. As the output of seppop is a list, we also use unlist() to get a simple vector.

Hobs <- lapply(seppop(myData), function(x) mean(summary(x)$Hobs))
Hobs <- unlist(Hobs)
Hobs

##      Pop1      Pop2      Pop3      Pop4      Pop5      Pop6      Pop7 
## 0.5511364 0.6413043 0.5446429 0.6250000 0.6573465 0.6691176 0.7500000

Again, we can store the results for observed and expected heterozygosity into a data.frame and compute the difference:

heteroz_per_pop <- data.frame(Hexp = Hs(myData), Hobs = Hobs)
heteroz_per_pop$diff <- heteroz_per_pop$Hexp - heteroz_per_pop$Hobs
heteroz_per_pop

##           Hexp      Hobs        diff
## Pop1 0.6747159 0.5511364  0.12357955
## Pop2 0.7258979 0.6413043  0.08459357
## Pop3 0.6246811 0.5446429  0.08003827
## Pop4 0.7206250 0.6250000  0.09562500
## Pop5 0.7556682 0.6573465  0.09832170
## Pop6 0.7480536 0.6691176  0.07893599
## Pop7 0.6854339 0.7500000 -0.06456612

We have already tested if these are significantly different above.

Geeky note:

To look at both the effect of the locus and the population at once, you can write some custom code:

heteroz_matrix <- do.call("cbind", lapply(seppop(myData),
                                          function(x) summary(x)$Hexp - summary(x)$Hobs))
heteroz_matrix

##              Pop1        Pop2         Pop3   Pop4        Pop5         Pop6        Pop7
## fca23 0.070247934  0.11720227  0.211734694 -0.170  0.04750000  0.190311419 -0.21074380
## fca43 0.094008264 -0.03402647  0.012755102  0.050  0.13875000  0.211072664 -0.22727273
## fca45 0.051652893 -0.03875236 -0.005102041 -0.040  0.14930556 -0.138408304 -0.11157025
## fca77 0.253099174  0.18336484  0.165816327  0.290  0.26250000 -0.126297578 -0.10743802
## fca78 0.237603306  0.16446125  0.257653061  0.320  0.03875000  0.008650519 -0.21074380
## fca90 0.001033058  0.11531191 -0.150510204 -0.035  0.06375000  0.262975779  0.12809917
## fca96 0.215909091  0.10396975  0.058673469  0.050 -0.06648199  0.186851211  0.17768595
## fca37 0.065082645  0.06521739  0.089285714  0.300  0.15250000  0.036332180  0.04545455

This can be easily plotted with lattice:

levelplot(heteroz_matrix, col.regions = viridis(100),
          main = "Heterozygosity: expected-observed",
          xlab = "Locus", ylab = "Population")

___

F-statistics

An effect of population subdivision on genetic variation is the reduction in observed heterozygotes. As we have seen previously (Wahlund effect).

The extent of reduction in observed heterozygotes can be used to quantify the level of differentiation between subpopulations.

This quantification is formalised in a series of hierarchical F-statistics.

–> We are using a measure of departure from HWE to quantify the extent of differentiation between populations.

F-statistics also provide a way to quantify the level of heterozygosity at the individual level, and if/how this departs from expectations of HWE in the population.

How are these linked?

F-statistics, from https://en.wikipedia.org/wiki/F-statistics

I = individual
S = sub-population
T = total-population

We will focus on the two most common F-statistics: $F_{IS}$ and $F_{ST}$.

$F_{IS}$ is known as the inbreeding coefficient.
- this essentially measures departures from random-mating.
- how does heterozygosity in individuals compare to that in the subpopulation?
- $F_{IS} = (H_S - H_I)/H_S$
$F_{ST}$ is known as the fixation index.
- measures the extent of genetic differentiation among subpopulations.
- how is heterozygosity in the subpopulation reduced relative to the total population.
- $F_{ST} = (H_T - H_S)/H_T$

Let’s calculate $F_{ST}$ for our previous example.

             AA    Aa    aa
      Pop1   50     0     0
      Pop2    0     0    50

Here, $p = 0.5$ and $q = 0.5$.

Our expected heterozygosity for the total population (ie. Pop1 and Pop2 together) is given by

$H_T = 2pq = 2 \times 0.5 \times 0.5 = 0.5$

As we have no variation in the subpopulations (ie. Pop1 or Pop2) $H_S = 0$ because $2pq = 2 \times 1 \times 0$

Then

$F_{ST} = (0.5 - 0) / 0.5 = 1.0$

Note: this has been a simple exploration of how F-statistics are calculated. This has been expanded upon, and not all software or R packages calculate F-stats in the same way. For example, some account for factors such as how individuals disperse (island model vs stepping-stone model), the mutation process (infinite alleles model vs step-wise mutation model) and other bias adjusters (e.g. taking into account sampling bias). It is thus important to understand and report which method you use.

To cite a package:

citation(package = "pegas")

## 
## To cite pegas in a publication use:
## 
##   Paradis E. 2010. pegas: an R package for population genetics with an integrated-modular approach. Bioinformatics 26: 419-420.
## 
## A BibTeX entry for LaTeX users is
## 
##   @Article{,
##     title = {pegas: an {R} package for population genetics with an integrated--modular approach},
##     author = {E. Paradis},
##     journal = {Bioinformatics},
##     year = {2010},
##     volume = {26},
##     pages = {419-420},
##   }
## 
## As pegas is evolving quickly, you may want to cite also its version number (found with 'library(help = pegas)').

packageVersion("pegas") ## don't forget to cite the version number!

## [1] '0.11'

Now let’s get back to myData.

We can use the Fst() function of the pegas package to calculate F-stats by locus, over all populations. But it requires the loci format, so we need to use as.loci() to transform our data.

Fst(as.loci(myData))

##              Fit        Fst         Fis
## fca23 0.18478245 0.07688651 0.116882646
## fca43 0.16430065 0.07043953 0.100973655
## fca45 0.08454681 0.07840534 0.006663954
## fca77 0.31145915 0.11736447 0.219903538
## fca78 0.23714930 0.04571881 0.200601766
## fca90 0.17238071 0.06492421 0.114917430
## fca96 0.22382373 0.05128694 0.181864038
## fca37 0.25929920 0.08156729 0.193516523

How would you interpret these values?

We can also build a data.frame with all the information we need:

data.frame(Hobs = summary(myData)$Hobs, Hexp = summary(myData)$Hexp, Fst(as.loci(myData))[, c("Fst", "Fis")])

##            Hobs      Hexp        Fst         Fis
## fca23 0.6666667 0.8044415 0.07688651 0.116882646
## fca43 0.6581197 0.7754767 0.07043953 0.100973655
## fca45 0.7155963 0.7687484 0.07840534 0.006663954
## fca77 0.6153846 0.8734020 0.11736447 0.219903538
## fca78 0.5982906 0.7748922 0.04571881 0.200601766
## fca90 0.6837607 0.8141939 0.06492421 0.114917430
## fca96 0.5948276 0.7565770 0.05128694 0.181864038
## fca37 0.4957265 0.6576813 0.08156729 0.193516523

Often we are also interested in specific $F_{ST}$ values between pairs of populations. Genetic differentiation among all populations does not need to be equal.

Let’s look at $F_{ST}$ between pops 1 and 2. We will extract the first two populations from myData and turn this into loci format using as.loci().

myData_pop12 <- as.loci(myData[pop = c("Pop1", "Pop2")])

Now we estimate $F_{ST}$ over all loci, obtained by using mean(), and constrain ourselves to the column of the output with the $F_{ST}$ values:

mean(Fst(myData_pop12)[, "Fst"])

## [1] 0.1031605

Now we have a measure of differentiation between Pop1 and Pop2.

This pairwise $F_{ST}$ value is the mean of $F_{ST}$ values of all loci. We can check $F_{ST}$ for every locus using Fst(myData_pop12)[, "Fst"].

As any estimate, the Fst is measured with some uncertainty. To represent this uncertainty, we can compute the 95$\%$ confidence interval of the mean Fst value using a “very simple” bootstrap (more advanced technics are possible):

res <- replicate(100, {
  myData_pop12 <- myData_pop12[, c(1, sample(attr(myData_pop12, "loci"), replace = TRUE))]
  mean(Fst(myData_pop12)[, "Fst"])
})
quantile(res, c(0.025, 0.975))

##       2.5%      97.5% 
## 0.08340952 0.12319740

hist(res, xlim = c(0, 0.2)); abline(v = 0, col = 2, lwd = 2, lty = 2)

Geeky note:

For now, there is no simple solution to compute pairwise Fst in R, but here is a solution using a home made function we prepared for you:

pairwise_F(myData, confint = FALSE) ## without Confidence Interval

##            Pop1       Pop2       Pop3       Pop4       Pop5       Pop6       Pop7
## Pop1         NA 0.10316054 0.11176328 0.07823261 0.08678415 0.11003071 0.08819144
## Pop2 0.10316054         NA 0.07680306 0.03648835 0.02176617 0.04722565 0.04794162
## Pop3 0.11176328 0.07680306         NA 0.05316041 0.08268424 0.11785210 0.11157795
## Pop4 0.07823261 0.03648835 0.05316041         NA 0.04222016 0.04885717 0.04768922
## Pop5 0.08678415 0.02176617 0.08268424 0.04222016         NA 0.04532272 0.04676997
## Pop6 0.11003071 0.04722565 0.11785210 0.04885717 0.04532272         NA 0.05978869
## Pop7 0.08819144 0.04794162 0.11157795 0.04768922 0.04676997 0.05978869         NA

pairwise_Fst <- pairwise_F(myData, confint = TRUE) ## with CI
lapply(pairwise_Fst, round, digit = 2) ## output after rounding with 2 digits

## $mean
##      Pop1 Pop2 Pop3 Pop4 Pop5 Pop6 Pop7
## Pop1   NA 0.10 0.11 0.08 0.09 0.11 0.09
## Pop2 0.10   NA 0.08 0.04 0.02 0.05 0.05
## Pop3 0.11 0.08   NA 0.05 0.08 0.12 0.11
## Pop4 0.08 0.04 0.05   NA 0.04 0.05 0.05
## Pop5 0.09 0.02 0.08 0.04   NA 0.05 0.05
## Pop6 0.11 0.05 0.12 0.05 0.05   NA 0.06
## Pop7 0.09 0.05 0.11 0.05 0.05 0.06   NA
## 
## $upr
##      Pop1 Pop2 Pop3 Pop4 Pop5 Pop6 Pop7
## Pop1   NA 0.12 0.15 0.10 0.10 0.13 0.11
## Pop2 0.12   NA 0.11 0.06 0.04 0.07 0.07
## Pop3 0.15 0.11   NA 0.08 0.13 0.14 0.15
## Pop4 0.10 0.06 0.08   NA 0.07 0.08 0.07
## Pop5 0.10 0.04 0.13 0.07   NA 0.06 0.09
## Pop6 0.13 0.07 0.14 0.08 0.06   NA 0.09
## Pop7 0.11 0.07 0.15 0.07 0.09 0.09   NA
## 
## $lwr
##      Pop1 Pop2 Pop3 Pop4 Pop5 Pop6 Pop7
## Pop1   NA 0.08 0.07 0.06 0.06 0.08 0.07
## Pop2 0.08   NA 0.04 0.01 0.00 0.01 0.02
## Pop3 0.07 0.04   NA 0.02 0.04 0.09 0.07
## Pop4 0.06 0.01 0.02   NA 0.01 0.01 0.03
## Pop5 0.06 0.00 0.04 0.01   NA 0.03 0.01
## Pop6 0.08 0.01 0.09 0.01 0.03   NA 0.02
## Pop7 0.07 0.02 0.07 0.03 0.01 0.02   NA

levelplot(pairwise_Fst$mean, col.regions = rev(grey.colors(30)))

Private alleles

It is worth checking if your populations have alleles that cannot be found in other populations. These are called “private alleles.”

We can use the private_alleles() function of poppr to get information about which alleles can only be found in a population.

Alleles are labelled as locus.allele.

private_alleles(myData)

##      fca23.148 fca43.143 fca77.132 fca78.138 fca90.205 fca90.189 fca90.181 fca96.101 fca96.121 fca37.192 fca37.194 fca37.200 fca37.218 fca37.212 fca37.186
## Pop1         0         0         0         0         0         0         0         1         1         3         2         0         0         0         0
## Pop2         0         0         0         0         0         0         0         0         0         0         0         1         1         0         0
## Pop3         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
## Pop4         0         1         0         2         0         0         0         0         0         0         0         0         0         2         1
## Pop5         2         0         1         0         1         1         0         0         0         0         0         0         0         0         0
## Pop6         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
## Pop7         0         0         0         0         0         0         2         0         0         0         0         0         0         0         0

apply(private_alleles(myData), 1, sum) ## number of private alleles per pop

## Pop1 Pop2 Pop3 Pop4 Pop5 Pop6 Pop7 
##    7    2    0    6    5    0    2

Private alleles will impact measures of differentiation between populations. Especially, if their frequency is high (i.e., if they are common).

plot(summary(myData)$pop.n.all ~ summary(myData)$n.by.pop, xlab = "Pop size", ylab = "Number of alleles", las = 1)

___

Trees & principal component analysis

Individual-based analyses frequently do not make assumptions about HWE, because they look at the differences between individuals.

But consider how this kind of information must be presented.

individual x individual

It is not surprising that we have come up with ways to visualize this kind of information.

Two very common ways are trees (or networks) and PCA.

One benefit of displaying information in this manner is that we can see if there are groupings of individuals that are more similar to eachother than they are to other individuals.

Proportion of shared alleles

A simple measure of genetic distance is the proportion of shared alleles. This measure does not make assumptions about mutation, genetic drift, etc.

It is, essentially, a measure of “how similar are two genotypes”, with a value ranging from 1 (= identical) to 0 (= no shared alleles).

adegenet has a function to calculate this: propShared().

The output of this function is a matrix with dimensions: no.of.samples x .no.of.samples. This is a bit too much to show on the screen if we consider the whole dataset of 117 samples!

Let us get the proportion of shared alleles among samples in Pop1 and display the information for the first 5 individuals:

similarity_mat <- propShared(myData[pop = "Pop1"])
similarity_mat[1:5, 1:5]

##          N7   N141   N142   N143   N144
## N7   1.0000 0.5000 0.5625 0.5000 0.5000
## N141 0.5000 1.0000 0.5000 0.4375 0.3750
## N142 0.5625 0.5000 1.0000 0.6250 0.5000
## N143 0.5000 0.4375 0.6250 1.0000 0.4375
## N144 0.5000 0.3750 0.5000 0.4375 1.0000

The function nj() of the ape package can create a neighbour joining (NJ) tree from a distance matrix.

We can use the similarity matrix we just computed to obtain a distance matrix and run the neighbour joining:

distance_mat <- 1 - similarity_mat
mynj <- nj(distance_mat)
mynj

## 
## Phylogenetic tree with 22 tips and 20 internal nodes.
## 
## Tip labels:
##  N7, N141, N142, N143, N144, N145, ...
## 
## Unrooted; includes branch lengths.

We have reconstructed a NJ tree based on the distance matrix. But we did not plot it!

For this we need to simply use the plot() function, which can handle the output of nj() (or rather, nj() was coded so that we can use plot()). We have several options for the type of tree to be plotted using type =, where we will set unrooted.

plot(mynj, type = "unrooted")

Other distance measures

adegenet and poppr offer us the ability to generate other distance measures. Most of the ones offered in poppr work for individuals (rather than populations).

Note: several of these distance measures make strong assumptions about the biological nature of our genetic samples.

Let’s try another measure that makes no assumptions: Prevosti’s distance, which we can use with prevosti.dist()

mynj_prevosti <- nj(prevosti.dist(seppop(myData)[["Pop1"]]))
plot(mynj_prevosti, type = "unrooted")

Note: we did not need to use 1 - in this case, because the function prevosti.dist() directly provide a distance.

Let us compare the two trees we just made to see if they match:

cophyloplot(mynj, mynj_prevosti,
            assoc =  cbind(mynj$tip.label, mynj_prevosti$tip.label))

Are you happy with that?

Now that we know how to make a NJ tree for a small dataset, we can do it for all samples in myData.

bignj <- nj(prevosti.dist(myData))
plot(bignj, type = "unrooted")

Geeky note: we can improve this plot by plotting colours and changing the type of tree. Here a possibility would be a “fan” plot:

plot(bignj, type = "fan", show.tip.label = FALSE, x.lim = c(-0.7, 0.7), no.margin = TRUE)
tiplabels(text = rownames(myData@tab),
          frame = "none",
          col = rainbow(nPop(myData))[as.numeric(myData@pop)], cex = 0.8, offset = 0.05)
legend("topleft", fill = rainbow(nPop(myData)),
       legend = popNames(myData), bty = "n",
       title = "Population")

Or perhaps a “radial” plot:

plot(bignj, type = "radial", show.tip.label = FALSE)
tiplabels(text = rownames(myData@tab),
          frame = "none",
          col = rainbow(nPop(myData))[as.numeric(myData@pop)], cex = 0.8, offset = 0.05)
legend("topleft", fill = rainbow(nPop(myData)),
       legend = popNames(myData), bty = "n",
       title = "Population")

PCA (Principal Component Analysis)

An alernative to neighbour joining is to reduce the dimensionality of the problem so that it can be ploted in 2 dimensions instead of one dimension per locus. Many technics exists but the most common one is the so-called Principal Component Analysis, or PCA for short.

This is how you run a PCA:

myData_matrix <- scaleGen(myData, center = FALSE, scale = FALSE, NA.method = "mean")
mypca <- dudi.pca(myData_matrix, center = TRUE, scale = FALSE, scannf = FALSE, nf = Inf)

What does the PCA do?

The PCA creates new dimensions…

head(mypca$li[, 1:4]) ## only show head for first 4 axes

##         Axis1      Axis2      Axis3      Axis4
## N7   3.768724 -1.0570158  0.3136979  0.1203990
## N141 3.103792  0.5957429 -1.0972396  0.3455816
## N142 2.802742 -1.8958561  0.7628661  0.6864441
## N143 3.420050 -0.6665903  0.4915670 -1.4854476
## N144 2.244425 -1.3172071  1.5874842  1.3820029
## N145 4.369164  1.9061879  0.0952608 -1.7985942

which are uncorrelated…

head(zapsmall(cor(mypca$li))[, 1:4])  ## only show head for first 4 axes

##       Axis1 Axis2 Axis3 Axis4
## Axis1     1     0     0     0
## Axis2     0     1     0     0
## Axis3     0     0     1     0
## Axis4     0     0     0     1
## Axis5     0     0     0     0
## Axis6     0     0     0     0

and which capture a decreasing amount of variation of the original loci:

barplot(mypca$eig,
        names.arg = colnames(mypca$li),
        cex.names = 0.5,
        col = heat.colors(length(mypca$eig)),
        las = 2, ylab = "Inertia")

or in cumulated percentage:

barplot(cumsum(100*mypca$eig/sum(mypca$eig)),
        names.arg = colnames(mypca$li),
        cex.names = 0.5,
        col = rev(heat.colors(length(mypca$eig))),
        las = 2, log = "y",
        ylab = "Cumulative proportion of variance explained")

Plotting the PCA

There are many ways to plot a PCA, but here we are interested in projecting the individuals into the new loci space, so we use the function s.class():

s.class(mypca$li, fac = pop(myData),
        col = rainbow(nPop(myData)), grid = FALSE, xax = 1, yax = 2, cpoint = 0)
s.label(mypca$li, add.plot = TRUE, boxes = FALSE, clabel = 0.5)
add.scatter.eig(mypca$eig[1:10], xax = 1, yax = 2, ratio = 0.15)

DCAPC (Discriminant Analysis of Principal Components)

The PCA does not force the group to be different, it just shows the overall structure. In contrast, the DAPC is a method that allows to explore whether some combinations of alleles would allows the characterisation of distinct groups.

Here is an example:

myclusters <- find.clusters(myData, n.clust = 3, n.pca = Inf) ## better not fixing n.clust and selecting it interactivelly!!!
dapc1 <- dapc(myData, pop = myclusters$grp, n.pca = Inf, n.da = Inf)
scatter(dapc1)

So yes, it seems that it is possible to create groups… but it is not clear what these groups correspond to in this dataset:

s.class(dapc1$ind.coord, fac = pop(myData),
        col = rainbow(nPop(myData)), grid = FALSE, xax = 1, yax = 2, cpoint = 1)

compoplot(dapc1, show.lab = TRUE, legend = FALSE, cex.names = 0.4,
          lab = paste(pop(myData), rownames(dapc1$tab)))

R for Genetics and Genomics Girls

Daniel Förster & Alexandre Courtiol

18 February, 2019

Before we really start

Why using R for genetics/genomics?

Outcome

Homework

About this document

Basic Markdown syntax

Basic knitr syntax

Getting all the R packages ready

Importing the data into R

The genepop format

Reading your data into R

R representations of your data

The genind format

The loci format

Checking your data

Missing data

MLG (MultiLocus Genotypes)

Ploidy

Are loci informative

Genetic variation

A very simple exploration

Allele vs genotype frequencies

Do you remember …

?

Assumptions

Freq. in next generation

Why do we care?

“Neutral” evolution

Testing the HWE

?

Wahlund

Linkage Disequilibrium

Descriptive statistics

Heterozygosity

F-statistics

Private alleles

Trees & principal component analysis

Proportion of shared alleles

Other distance measures

PCA (Principal Component Analysis)

What does the PCA do?

Plotting the PCA

DCAPC (Discriminant Analysis of Principal Components)

THE END

Why using `R` for genetics/genomics?

Basic `knitr` syntax

Getting all the `R` packages ready

Importing the data into `R`

The `genepop` format

Reading your data into `R`

`R` representations of your data

The `genind` format

The `loci` format